Question

what mathematic properties should i use for these equations
1. 2(3-7x) + 4x=8-5x
2. 6-14x+4x=8-5x
3. 6-10x=8-5x
4. 6-5x=8
5. 5x=2
6. x= -2/5

Answer 1

To solve the equations you've provided, you can use several mathematical properties, including the distributive property, the properties of equality, and the addition and multiplication properties. Here’s a summary of the properties you might use for each equation:

1. **Distributive Property**: Use this to simplify the left side:
\[
2(3 - 7x) + 4x = 8 - 5x
\]
will become
\[
6 - 14x + 4x = 8 - 5x.
\]

2. **Combining Like Terms**: In this equation, you can combine like terms on both sides:
\[
6 - 14x + 4x = 8 - 5x
\]
simplifies to
\[
6 - 10x = 8 - 5x.
\]

3. **Moving Terms**: You can use the properties of equality to move terms to one side:
\[
6 - 10x = 8 - 5x
\]
can be rearranged to isolate \( x \).

4. **Subtracting from Both Sides**: You can subtract 6 from both sides:
\[
6 - 5x = 8
\]
becomes
\[
-5x = 8 - 6.
\]

5. **Dividing by a Constant**: To isolate \( x \):
\[
5x = 2
\]
can be solved by dividing both sides by 5.

6. **Understanding Negative Values**: This step shows a simple calculation of division:
\[
x = -\frac{2}{5}.
\]

In summary, the properties you would use include:

- **Distributive Property** for expanding expressions.
- **Combining Like Terms** for simplifying expressions.
- **Properties of Equality** (addition and subtraction) for manipulating equations to isolate variables.
- **Division Property of Equality** for solving for the variable.

By applying these properties step-by-step, you can solve each equation systematically.